New directions in inversesemigroup theory
Mark V Lawson
Heriot-Watt University, Edinburgh
June 2016
Celebrating the LXth birthday of
Jorge Almeida and Gracinda Gomes
1
With the collaboration of
Peter Hines, Anja Kudryavtseva, Johannes Kel-
lendonk, Daniel Lenz, Stuart Margolis, Pedro
Resende, Phil Scott, Ben Steinberg and Alis-
tair Wallis.
and developing ideas to be found in
(amongst others)
Jean Renault, Alexander Kumjian, Alan Pater-
son, Ruy Exel, Hiroki Matui, . . .
2
Idea
Develop inverse semigroup theory as
the abstract theory of pseudogroups
of transformations.
It is worth adding that much if not all ‘classi-
cal’ inverse semigroup theory will be used. For
example, fundamental inverse semigroups and
the Munn semigroup play an important role in
this approach.
3
1. Pseudogroups of transformations
Let X be a topological space. A pseudogroup
of transformations on X is a collection Γ of
homeomorphisms between the open subsets of
X (called partial homeomorphisms) such that
1. Γ is closed under composition.
2. Γ is closed under ‘inverses’.
3. Γ contains all the identity functions on the
open subsets.
4. Γ is closed under arbitrary non-empty unions
when those unions are partial bijections.
4
• Pseudogroups are important in the foun-
dations of geometry.
• The idempotents in Γ are precisely the iden-
tity functions on the open subsets of the
topological space. They form a complete,
infinitely distributive lattice or frame.
• Johnstone on the origins of frame theory
“It was Ehresmann . . . and his stu-
dent Benabou . . . who first took the
decisive step in regarding complete
Heyting algebras as ‘generalized topo-
logical spaces”.
However, Johnstone does not say why Ehres-
mann was led to his frame-theoretic view-
point of topological spaces. The reason
was pseudogroups.
5
• Pseudogroups are usually replaced by their
groupoids of germs but pseudogroups nev-
ertheless persist.
• The algebraic part of pseudogroup theory
became inverse semigroup theory.
6
But recent developments show that it is fruit-
ful to bring these divergent approaches back
together.
In particular, inverse semigroups and frames.
This is in the spirit of Ehresmann’s work.
7
2. Inverse semigroups
Inverse semigroups arose by abstracting pseu-
dogroups of transformations in the same way
that groups arose by abstracting groups of trans-
formations.
There were three independent approaches:
1. Charles Ehresmann (1905–1979) in France.
2. Gordon B. Preston (1925–2015) in the UK.
3. Viktor V. Wagner (1908–1981) in the USSR.
They all three converge on the definition of
‘inverse semigroup’.
8
Revision
A semigroup S is said to be inverse if for each
a ∈ S there exists a unique element a−1 such
that a = aa−1a and a−1 = a−1aa−1.
The idempotents in an inverse semigroup com-
mute with each other. We speak of the semi-
lattice of idempotents E(S) of the inverse semi-
group S.
Pseudogroups of transformations are inverse
semigroups BUT they also have order-completeness
properties.
9
Let S be an inverse semigroup. Define a ≤ b if
a = ba−1a.
The relation ≤ is a partial order with respect to
which S is a partially ordered semigroup called
the natural partial order.
Suppose that a, b ≤ c. Then ab−1 ≤ cc−1 and
a−1b ≤ c−1c. Thus a necessary condition for a
and b to have an upper bound is that a−1b and
ab−1 be idempotent.
Define a ∼ b if a−1b and ab−1 are idempotent.
This is the compatibility relation.
A subset is said to be compatible if each pair
of distinct elements in the set is compatible.
10
3. Pseudogroups
An inverse semigroup is said to have finite (resp.
infinite) joins if each finite (resp. arbitrary)
compatible subset has a join.
Definition. An inverse monoid is said to be a
pseudogroup if it has infinite joins and multi-
plication distributes over such joins.
Theorem [Schein completion] Let S be an in-
verse semigroup. Then there is a pseudogroup
Γ(S) and a map γ : S → Γ(S) universal for
maps to pseudogroups.
Pseudogroups are the correct abstractions of
pseudogroups of transformations, but their order-
completeness properties make them look spe-
cial.11
Let S be a pseudogroup. An element a ∈ S
is said to be finite if a ≤∨i∈I ai implies that
a ≤∨ni∈1 ai for some finite subset 1, . . . , n ⊆ I.
Denote by K(S) the set of finite elements of S.
An inverse semigroup is said to be distribu-tive if it has finite joins and multiplication dis-tributes over such joins.
A pseudogroup S is said to be coherent if eachelement of S is a join of finite elements andthe set of finite elements forms a distributiveinverse semigroup.
Theorem The category of distributive inversesemigroups is equivalent to the category of co-herent pseudogroups.
A distributive inverse semigroup is said to beBoolean if its semilattice of idempotents formsa (generalized) Boolean algebra.
12
4. Non-commutative frame theory
Commutative Non-commutative
frame pseudogroup
distributive lattice distributive inverse semigroup
Boolean algebra Boolean inverse semigroup
We are therefore led to view inverse semigroup
theory as non-commutative frame theory:
• Meet semilattices −→ inverse semigroups.
• Frames −→ pseudogroups.
Remark There is also a connection with quan-
tales. This is discussed in P. Resende, Etale
groupoids and their quantales, Adv. Math.
208 (2007), 147–209.
13
5. An example
For n ≥ 2, define the inverse semigroup Pn by
the monoid-with-zero presentation
Pn = 〈a1, . . . , an, a−11 , . . . , a−1
n : a−1i aj = δij〉,
the polycyclic inverse monoid on n generators.
This is an aperiodic inverse monoid. In partic-
ular, its group of units is trivial.
14
Define Cn to be the distributive inverse monoid
generated by Pn together with the ‘relation’
1 =n∨i=1
aia−1i .
This was done using a bare-hands method in:
The polycyclic monoids Pn and the Thomp-
son groups Vn,1, Communications in algebra
35 (2007), 4068–4087.
In fact, the Cn are countable atomless Boolean
inverse monoids called the Cuntz inverse monoids,
whose groups of units are the Thompson groups
V2, V3, . . ., respectively.
They are analogues of On, the Cuntz C∗-algebras.
Representations of the inverse monoids Cn are
(unwittingly) the subject of Iterated function
systems and permutation representations of the
Cuntz algebra by O. Bratteli and P. E. T. Jor-
gensen, AMS, 1999.
15
What is going on here is that elements of Pnare being glued together in suitable ways to
yield elements of Cn.
This leads to (many) invertible elements being
constructed.
Thus an inverse semigroup together with ‘suit-
able relations’ can be used to construct new
inverse semigroups with interesting properties.
16
6. Coverages (sketch)
• Informally, a coverage T on an inverse semi-
group S is a collection of ‘abstract rela-
tions’ interpreted to mean a =∨i∈I ai.
• Pseudogroup = inverse semigroup + coverage.
• If the pseduogroup is coherent, we actually havedistributive inverse semigroup = inverse semi-group + coverage.
• In special cases, we may haveBoolean inverse semigroup = inverse semigroup+ coverage.
17
7. Non-commutative Stone dualities
The classical theory of pseudogroups of trans-
formations requires topology.
We generalize the classical connection between
topological spaces and frames, which we now
recall.
To each topological space X there is the asso-
ciated frame of open sets Ω(X).
To each frame L there is the associated topo-
logical space of completely prime filters Sp(L).
18
The following is classical.
Theorem The functor L 7→ Sp(L) from the
dual of the category of frames to the category
of spaces is right adjoint to the functor X 7→Ω(X).
A frame is called spatial if elements can be
distinguished by means of completely prime fil-
ters.
A space is called sober if points and completely
prime filters are in bijective correspondence.
19
We replace topological spaces (commutative)
by topological groupoids (non-commutative).
We view categories as 1-sorted structures (over
sets): everything is an arrow. Objects are iden-
tified with identity arrows.
A groupoid is a category in which every arrow
is invertible.
We regard groupoids as ‘groups with many
identities’. The set of identities is Go.
Key definition. Let G be a groupoid with set
of identities Go. A subset A ⊆ G is called a
local bisection if A−1A,AA−1 ⊆ Go. We say
that S is a bisection if A−1A = AA−1 = Go.
Proposition The set of all local bisections
of a groupoid forms a Boolean inverse meet-
monoid.20
A topological groupoid is said to be etale if its
domain and range maps are local homeomor-
phisms.
Why etale? This is explained by the following
result.
Theorem [Resende] A topological groupoid is
etale if and only if its set of open subsets forms
a monoid under multiplication of subsets.
Etale groupoids therefore have a strong alge-
braic character.
21
There are two basic constructions.
• Let G be an etale groupoid. Denote by
B(G) the set of all open local bisections of
G. Then B(G) is a pseudogroup.
• Let S be a pseudogroup. Denote by G(S)
the set of all completely prime filters of S.
Then G(S) is an etale groupoid. [This is
the ‘hard’ direction].
22
Denote by Ps a suitable category of pseudogroups
and by Etale a suitable category of etale groupoids.
Theorem [The main adjunction] The functor
G : Psop → Etale is right adjoint to the functor
B : Etale→ Psop.
Theorem [The main equivalence] There is a
dual equivalence between the category of spa-
tial pseudogroups and the category of sober
etale groupoids.
23
We now restrict to coherent pseudogroups. Theseare automatically spatial.
An etale groupoid is said to be spectral if itsidentity space is sober, has a basis of compact-open sets and if the intersection of any twosuch compact-open sets is compact-open. Werefer to spectral groupoids rather than spectraletale groupoids.
Theorem There is a dual equivalence betweenthe category of distributive inverse semigroupsand the category of spectral groupoids.
• Under this duality, a spectral groupoid G
is mapped to the set of all compact-openlocal bisections KB(G)
• Under this duality, a distributive inverse semi-group is mapped to the set of all primefilters GP (S).
24
An etale groupoid is said to be Boolean if its
identity space is Hausdorff, locally compact
and has a basis of clopen sets. We refer to
Boolean groupoids rather than Boolean etale
groupoids.
Proposition A distributive inverse semigroup
is Boolean if and only if prime filters and ul-
trafilters are the same.
Theorem There is a dual equivalence between
the category of Boolean inverse semigroups
and the category of Boolean groupoids.
Theorem There is a dual equivalence between
the category of Boolean inverse meet-semigroups
and the category of Haudorff Boolean groupoids.
25
Algebra Topology
Semigroup Locally compact
Monoid Compact
Meet-semigroup Hausdorff
The theory discussed so far can be found in:
M. V. Lawson, D.H. Lenz, Pseudogroups and
their etale groupoids, Adv. Math. 244 (2013),
117–170.
26
8. Boolean inverse semigroups
These have proved to be the most interesting
class of inverse semigroups viewed from this
perspective. They are genuine non-commutative
generalizations of Boolean algebras.
Example Let
S0τ0→ S1
τ1→ S2τ2→ . . .
be a sequence of Boolean inverse ∧-monoids
and injective morphisms. Then the direct limit
lim−→Si is a Boolean inverse ∧-monoid. If the Siare finite direct products of finite symmetric
inverse monoids then the direct limit is called
an AF inverse monoid. There is a close con-
nection between such inverse monoids and MV
algebras (another generalization of Boolean al-
gebras).
27
Let S be an inverse semigroup. An ideal I in
S is a non-empty subset such that SIS ⊆ I.
Now let S be a Boolean inverse semigroup. A
∨-ideal I in S is an ideal with the additional
property that it is closed under finite compat-
ible joins.
A Boolean inverse semigroup is said to be 0-
simplifying if it has no non-trivial ∨-ideals.
Key definition. A Boolean inverse semigroup
that is both fundamental and 0-simplifying is
said to be simple.
Caution! Simple in this context means what
is defined above.
28
Theorem
1. The finite Boolean inverse semigroups are
isomorphic to the set of all local bisections
of a finite discrete groupoid.
2. The finite fundamental Boolean inverse semi-
groups are isomorphic to finite direct prod-
ucts of finite symmetric inverse monoids.
3. The finite simple Boolean inverse semigroups
are isomorphic to finite direct products of
finite symmetric inverse monoids.
29
Theorem [The simple alternative] A simple
Boolean inverse monoid is either isomorphic to
a finite symmetric inverse monoid or atomless.
Under classical Stone duality, the Cantor space
corresponds to the (unique) countable atom-
less Boolean algebra; it is convenient to give
this a name and we shall refer to it as the
Tarski algebra.
Corollary A simple countable Boolean inverse
monoid has the Tarski algebra as its set of
idempotents.
30
A non-zero element a in an inverse semigroup
is said to be an infinitesimal if a2 = 0. The
following result explains why infinitesimals are
important.
Proposition Let S be a Boolean inverse monoid
and let a be an infinitesimal. Then
a ∨ a−1 ∨ (a−1a ∨ aa−1)
is an involution.
We call an involution that arises in this way a
transposition.
31
Programme
The subgroup of the group of
units of a simple Boolean inverse
monoid generated by the transposi-
tions should be regarded as a gener-
alization of a finite symmetric group.
This is the theme of ongoing work by
Hiroki Matui and Volodymyr Nekra-
shevych.
32
Theorem [Wehrung] Boolean inverse semigroupsform a variety with respect to a suitable signa-ture; this variety is congruence-permutable.
Programme
Study varieties of Boolean inverse
semigroups.
Theorem [Paterson, Wehrung] Let S be an in-verse subsemigroup of the multiplicative semi-group of a C∗-algebra R in such as way thatthe inverse in S is the involution of R. Thenthere is a Boolean inverse semigroup B suchthat S ⊆ B ⊆ R such that the inverse in B isthe involution in R.
Programme
Investigate the relationship between
Boolean inverse monoids and C∗-algebras of real rank zero. The Cuntz
inverse monoids Cn and the AF in-
verse monoids are good examples.
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9. Refinements
Recall that every groupoid is a union of its
connected components.
A subset of a groupoid that is a union of con-
nected components is said to be invariant.
An etale groupoid is said to be minimal if there
are no non-trivial open invariant subsets.
Theorem Under non-commutative Stone du-
ality, 0-simplifying Boolean inverse semigroups
correspond to minimal Boolean groupoids.
34
Let G be an etale groupoid.
Its isotropy subgroupoid Iso(G) is the subgroupoid
consisting of the union of its local groups.
The groupoid G is said to be effective if the
interior of Iso(G), denoted by Iso(G), is equal
to the space of identities of G.
Let S be an inverse semigroup. It is fundamen-
tal if the only elements that commute with all
idempotents are idempotents.
35
Theorem Under the dual equivalences.
1. Fundamental spatial pseudogroups corre-
spond to effective sober etale groupoids.
2. Fundamental distributive inverse semigroups
correspond to effective spectral groupoids.
3. Fundamental Boolean inverse semigroups
correspond to effective Boolean groupoids.
36
Definition Denote by Homeo(S ) the group of
homeomorphisms of the Boolean space S . By
a Boolean full group, we mean a subgroup G of
Homeo(S ) satisfying the following condition:
let e1, . . . , en be a finite partition of S by
clopen sets and let g1, . . . , gn be a finite subset
of G such that g1e1, . . . , gnen is a partition
of S also by clopen sets. Then the union of
the partial bijections (g1 | e1), . . . , (gn | en) is
an element of G. We call this property fullness
and term full those subgroups of Homeo(S )
that satisfy this property.
37
Theorem The following three classes of struc-
ture are equivalent.
1. Minimal Boolean full groups.
2. Simple Boolean inverse monoids
3. Minimal, effective Boolean groupoids.
38
Let S be a compact Hausdorff space. If α ∈Homeo(S ), define
supp(α) = clx ∈ S : α(x) 6= x
the support of α.
Theorem The following three classes of struc-
ture are equivalent.
1. Minimal Boolean full groups in which each
element has clopen support.
2. Simple Boolean inverse meet-monoids
3. Minimal, effective, Hausdorff Boolean groupoids.
39
Theorem The following three classes of struc-
ture are equivalent.
1. Minimal Boolean full groups in which each
element has a clopen fixed-point set.
2. Simple basic Boolean inverse meet-monoids
3. Minimal, effective, Hausdorff, principal Boolean
groupoids.
40
Let S be a Boolean inverse monoid. Denote
by Sym(S) the subgroup of the group of units
of S generated by transpositions.
Let a and b be infinitesimals and put c = (ba)−1
as in the following diagram
e2b||
e3 c// e1
abb
where the idempotents e1, e2, e3 are mutually
orthogonal. Put e = e1∨e2∨e3. Then a∨b∨c∨eis a unit called a 3-cycle.
Denote by Alt(S) the subgroup of the group of
units of S generated by 3-cycles.
Theorem [Nekrashevych] Let S be a simple
Boolean inverse monoid. Then Alt(S) is simple.
41
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